Standard voting power indexes do not work: An empirical analysis

British Journal of Political Science

Volumn 34, Issue 4, 2004, Pages 657-674

  Gelman, Andrew ^a,b   Katz, Jonathan N ^c   Bafumi, Joseph ^a  

^a Och Spine at New York Presbyterian Hospitals   (United States)

^b Department of Statistics ^*  (United States)

^c CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 7644225269     PISSN: 00071234     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0007123404000237     Document Type: Article

References (75)

1. For example, see D. S. Felsenthal and M. Machover, 'Enlargement of the EU and Weighted Voting in Its Council of Ministers', Report VPP 01/00 (2000), Centre for Philosophy of Natural and Social Science, London School of Economics and Political Science; D. Leech, 'Designing the Voting System for the Council of the European Union', Public Choice, 113 (2002), 437-64; and D. Leech, 'Voting Power in the Governance of the IMF', Annals of Operations Research, 109 (2002), 373-95.
2. For example, see D. S. Felsenthal and M. Machover, 'Enlargement of the EU and Weighted Voting in Its Council of Ministers', Report VPP 01/00 (2000), Centre for Philosophy of Natural and Social Science, London School of Economics and Political Science; D. Leech, 'Designing the Voting System for the Council of the European Union', Public Choice, 113 (2002), 437-64; and D. Leech, 'Voting Power in the Governance of the IMF', Annals of Operations Research, 109 (2002), 373-95.
3. For example, see D. S. Felsenthal and M. Machover, 'Enlargement of the EU and Weighted Voting in Its Council of Ministers', Report VPP 01/00 (2000), Centre for Philosophy of Natural and Social Science, London School of Economics and Political Science; D. Leech, 'Designing the Voting System for the Council of the European Union', Public Choice, 113 (2002), 437-64; and D. Leech, 'Voting Power in the Governance of the IMF', Annals of Operations Research, 109 (2002), 373-95.
4. A complete history is beyond the scope of this article, but see D. S. Felsenthal and M. Machover, The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes (Northampton, Mass.: Edward Elgar, 1998), for a brief history.
5. See, for example, Felsenthal and Machover, The Measurement of Voting Power, and D. G. Saari and K. K. Sieberg, 'Some Surprising Properties of Power Indices', Games and Economic Behavior, 36 (2000), 241-63.
6. See, for example, Felsenthal and Machover, The Measurement of Voting Power, and D. G. Saari and K. K. Sieberg, 'Some Surprising Properties of Power Indices', Games and Economic Behavior, 36 (2000), 241-63.
7. J. R. Banzhaf, 'One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College', Villanova Law Review, 16 (1968), 304-32.
8. Felsenthal and Machover, 'Enlargement of the EU'. See also L. S. Penrose, 'The Elementary Statistics of Majority Voting', Journal of the Royal Statistical Society 109 (1946), 53-7.
9. Felsenthal and Machover, 'Enlargement of the EU'. See also L. S. Penrose, 'The Elementary Statistics of Majority Voting', Journal of the Royal Statistical Society 109 (1946), 53-7.
10. As is noted by A. D. Heard and T. B. Swartz, 'Extended Voting Measures', Canadian Journal of Statistics, 27 (1999), 177-86.
11. J. M. Snyder, M. M. Ting and S. Ansolabehere, 'Legislative Bargaining Under Weighted Voting' (Technical report, Department of Economics, Massachusetts Institute of Technology, 2001).
12. For example, see G. Garrett, and G. Tsebelis, 'Why Resist the Temptation to Apply Power Indices to the European Union?' Journal of Theoretical Politics, 11 (1999), 291-308.
13. For a more thorough discussion, see Felsenthal and Machover, The Measurement of Voting Power.
14. See Felsenthal and Machover, The Measurement of Voting Power, for complete discussion of the types of voting power indexes.
15. j) in the larger council. The voting power of an individual citizen has the same two-stage mathematical structure (as described in this section) whether the aggregation is by representatives as in the Council of Ministers or (nearly) automatic as in the Electoral College.
16. See Heard and Schwartz, 'Extended Voting Measures', and Felsenthal and Machover, The Measurement of Voting Power.
17. See Heard and Schwartz, 'Extended Voting Measures', and Felsenthal and Machover, The Measurement of Voting Power.
18. W. H. Riker and P. C. Ordeshook, 'A Theory of the Calculus of Voting', American Political Science Review, 62 (1968), 25-42; J. Ferejohn and M. Fiorina, 'The Paradox of Not Voting: A Decision Theoretic Analysis', American Political Science Review, 68 (1974), 525-36; S. J. Brams and M. D. Davis, 'The 3/2's Rule in Presidential Campaigning', American Political Science Review, 68 (1974), 113-34; and J. H. Aldrich, 'Rational Choice and Turnout', American Journal of Political Science, 37 (1993), 246-78.
19. W. H. Riker and P. C. Ordeshook, 'A Theory of the Calculus of Voting', American Political Science Review, 62 (1968), 25-42; J. Ferejohn and M. Fiorina, 'The Paradox of Not Voting: A Decision Theoretic Analysis', American Political Science Review, 68 (1974), 525-36; S. J. Brams and M. D. Davis, 'The 3/2's Rule in Presidential Campaigning', American Political Science Review, 68 (1974), 113-34; and J. H. Aldrich, 'Rational Choice and Turnout', American Journal of Political Science, 37 (1993), 246-78.
20. W. H. Riker and P. C. Ordeshook, 'A Theory of the Calculus of Voting', American Political Science Review, 62 (1968), 25-42; J. Ferejohn and M. Fiorina, 'The Paradox of Not Voting: A Decision Theoretic Analysis', American Political Science Review, 68 (1974), 525-36; S. J. Brams and M. D. Davis, 'The 3/2's Rule in Presidential Campaigning', American Political Science Review, 68 (1974), 113-34; and J. H. Aldrich, 'Rational Choice and Turnout', American Journal of Political Science, 37 (1993), 246-78.
21. W. H. Riker and P. C. Ordeshook, 'A Theory of the Calculus of Voting', American Political Science Review, 62 (1968), 25-42; J. Ferejohn and M. Fiorina, 'The Paradox of Not Voting: A Decision Theoretic Analysis', American Political Science Review, 68 (1974), 525-36; S. J. Brams and M. D. Davis, 'The 3/2's Rule in Presidential Campaigning', American Political Science Review, 68 (1974), 113-34; and J. H. Aldrich, 'Rational Choice and Turnout', American Journal of Political Science, 37 (1993), 246-78.
22. J. R. Banzhaf, 'Weighted Voting Doesn't Work: A Mathematical Analysis', Rutgers Law Review, 19 (1965), 317-43. The index had also been proposed by L. S. Penrose, 'The Elementary Statistics of Majority Voting', Journal of the Royal Statistical Society, 109 (1946), 53-7, and others; see Felsenthal and Machover, 'Enlargement of the EU'. Section 1.2.3, for a historical overview.
23. J. R. Banzhaf, 'Weighted Voting Doesn't Work: A Mathematical Analysis', Rutgers Law Review, 19 (1965), 317-43. The index had also been proposed by L. S. Penrose, 'The Elementary Statistics of Majority Voting', Journal of the Royal Statistical Society, 109 (1946), 53-7, and others; see Felsenthal and Machover, 'Enlargement of the EU'. Section 1.2.3, for a historical overview.
24. J. R. Banzhaf, 'Weighted Voting Doesn't Work: A Mathematical Analysis', Rutgers Law Review, 19 (1965), 317-43. The index had also been proposed by L. S. Penrose, 'The Elementary Statistics of Majority Voting', Journal of the Royal Statistical Society, 109 (1946), 53-7, and others; see Felsenthal and Machover, 'Enlargement of the EU'. Section 1.2.3, for a historical overview.
25. Some examples of this phenomenon in actual voting systems are provided by Felsenthal and Machover, The Measurement of Voting Power, and D. Leech, 'The Utility of the Voting Power Approach', part of a forum of responses by Felsenthal and others, 'In Defence of Voting Power Analysis', European Union Politics 4 (2003).
26. Some examples of this phenomenon in actual voting systems are provided by Felsenthal and Machover, The Measurement of Voting Power, and D. Leech, 'The Utility of the Voting Power Approach', part of a forum of responses by Felsenthal and others, 'In Defence of Voting Power Analysis', European Union Politics 4 (2003).
27. Some examples of this phenomenon in actual voting systems are provided by Felsenthal and Machover, The Measurement of Voting Power, and D. Leech, 'The Utility of the Voting Power Approach', part of a forum of responses by Felsenthal and others, 'In Defence of Voting Power Analysis', European Union Politics 4 (2003).
28. This would be most simply done using the normal distribution, but other forms are possible, such as the scaled beta distribution used by A. Gelman, G. King, and W.J. Boscardin, 'Estimating the Probability of Events That Have Never Occurred: When Is Your Vote Decisive?' Journal of the American Statistical Association, 93 (1998), 1-9.
29. Banzhaf, 'One Man, 3.312 Votes'; Brams and Davis, 'The 3/2's Rule in Presidential Campaigning'.
30. Banzhaf, 'One Man, 3.312 Votes'; Brams and Davis, 'The 3/2's Rule in Presidential Campaigning'.
31. Penrose, 'The Elementary Statistics of Majority Voting', and Felsenthal and Machover, 'Enlargement of the EU'.
32. Penrose, 'The Elementary Statistics of Majority Voting', and Felsenthal and Machover, 'Enlargement of the EU'.
33. See, for example, N. Beck, 'A Note on the Probability of a Tied Election', Public Choice, 23 (1975), 75-9; H. Margolis, 'Probability of a Tie Election', Public Choice, 31 (1977), 134-7; S. Merrill, 'Citizen Voting Power under the Electoral College: A Stochastic Model Based on State Voting Patterns', SIAM Journal of Applied Mathematics, 34 (1978), 376-90; and G. Chamberlain and M. Rothchild, 'A Note on the Probability of Casting a Decisive Vote', Journal of Economic Theory, 25 (1981), 152-62.
34. See, for example, N. Beck, 'A Note on the Probability of a Tied Election', Public Choice, 23 (1975), 75-9; H. Margolis, 'Probability of a Tie Election', Public Choice, 31 (1977), 134-7; S. Merrill, 'Citizen Voting Power under the Electoral College: A Stochastic Model Based on State Voting Patterns', SIAM Journal of Applied Mathematics, 34 (1978), 376-90; and G. Chamberlain and M. Rothchild, 'A Note on the Probability of Casting a Decisive Vote', Journal of Economic Theory, 25 (1981), 152-62.
35. See, for example, N. Beck, 'A Note on the Probability of a Tied Election', Public Choice, 23 (1975), 75-9; H. Margolis, 'Probability of a Tie Election', Public Choice, 31 (1977), 134-7; S. Merrill, 'Citizen Voting Power under the Electoral College: A Stochastic Model Based on State Voting Patterns', SIAM Journal of Applied Mathematics, 34 (1978), 376-90; and G. Chamberlain and M. Rothchild, 'A Note on the Probability of Casting a Decisive Vote', Journal of Economic Theory, 25 (1981), 152-62.
36. See, for example, N. Beck, 'A Note on the Probability of a Tied Election', Public Choice, 23 (1975), 75-9; H. Margolis, 'Probability of a Tie Election', Public Choice, 31 (1977), 134-7; S. Merrill, 'Citizen Voting Power under the Electoral College: A Stochastic Model Based on State Voting Patterns', SIAM Journal of Applied Mathematics, 34 (1978), 376-90; and G. Chamberlain and M. Rothchild, 'A Note on the Probability of Casting a Decisive Vote', Journal of Economic Theory, 25 (1981), 152-62.
37. See, for example, J. E. Campbell, 'Forecasting the Presidential Vote in the States', American Journal of Political Science, 36 (1992), 386-407.
38. Gelman, King and Boscardin, 'Estimating the Probability of Events That Have Never Occurred'.
39. See A. Gelman and G. King, 'A Unified Model for Evaluating Electoral Systems and Redistricting Plans', American Journal of Political Science, 38 (1994), 514-54.
40. I. J. Good and L. S. Mayer, 'Estimating the Efficacy of a Vote', Behavioral Science, 20 (1975), 25-33; Chamberlain and Rothchild, 'A Note on the Probability of Casting a Decisive Vote'.
41. I. J. Good and L. S. Mayer, 'Estimating the Efficacy of a Vote', Behavioral Science, 20 (1975), 25-33; Chamberlain and Rothchild, 'A Note on the Probability of Casting a Decisive Vote'.
42. Penrose, 'The Elementary Statistics of Majority Voting', and Felsenthal and Machover, 'Enlargement of the EU', Section 2.3.
43. Penrose, 'The Elementary Statistics of Majority Voting', and Felsenthal and Machover, 'Enlargement of the EU', Section 2.3.
44. Banzhaf, 'One Man, 3.312 Votes'.
45. Brahms and Davis, 'The 3/2's Rule in Presidential Campaigning'.
46. I. Mann and L. S. Shapley, 'The A Priori Voting Strength of the Electoral College', RAND memo (1960) reprinted in M. Shubik, ed., Game Theory and Related Approaches to Social Behavior (New York: Wiley, 1964); G. Owen, 'Evaluation of a Presidential Election Game', American Political Science Review, 69 (1975), 947-53; G. Rabinowitz and S. E. Macdonald, 'The Power of the States in U.S. Presidential Elections', American Political Science Review, 80 (1986), 65-87.
47. I. Mann and L. S. Shapley, 'The A Priori Voting Strength of the Electoral College', RAND memo (1960) reprinted in M. Shubik, ed., Game Theory and Related Approaches to Social Behavior (New York: Wiley, 1964); G. Owen, 'Evaluation of a Presidential Election Game', American Political Science Review, 69 (1975), 947-53; G. Rabinowitz and S. E. Macdonald, 'The Power of the States in U.S. Presidential Elections', American Political Science Review, 80 (1986), 65-87.
48. I. Mann and L. S. Shapley, 'The A Priori Voting Strength of the Electoral College', RAND memo (1960) reprinted in M. Shubik, ed., Game Theory and Related Approaches to Social Behavior (New York: Wiley, 1964); G. Owen, 'Evaluation of a Presidential Election Game', American Political Science Review, 69 (1975), 947-53; G. Rabinowitz and S. E. Macdonald, 'The Power of the States in U.S. Presidential Elections', American Political Science Review, 80 (1986), 65-87.
49. T. Noah, 'Faithless elector watch: gimme "equal protection"', Slate, 13 December 2000, http://slate.msn.com/?id=1006680
50. Banzhaf, 'One Man, 3.312 Votes'; Brams and Davis, 'The 3/2's Rule in Presidential Campaigning', and S. J. Brams and M. D. Davis, 'Comment on "Campaign Resource Allocation under the Electoral College"' by C. S. Colantoni, T. J. Levesque and P. C. Ordeshook, American Political Science Review, 69 (1975), 155-6; Owen, 'Evaluation of a Presidential Election Game'.
51. Banzhaf, 'One Man, 3.312 Votes'; Brams and Davis, 'The 3/2's Rule in Presidential Campaigning', and S. J. Brams and M. D. Davis, 'Comment on "Campaign Resource Allocation under the Electoral College"' by C. S. Colantoni, T. J. Levesque and P. C. Ordeshook, American Political Science Review, 69 (1975), 155-6; Owen, 'Evaluation of a Presidential Election Game'.
52. Banzhaf, 'One Man, 3.312 Votes'; Brams and Davis, 'The 3/2's Rule in Presidential Campaigning', and S. J. Brams and M. D. Davis, 'Comment on "Campaign Resource Allocation under the Electoral College"' by C. S. Colantoni, T. J. Levesque and P. C. Ordeshook, American Political Science Review, 69 (1975), 155-6; Owen, 'Evaluation of a Presidential Election Game'.
53. Banzhaf, 'One Man, 3.312 Votes'; Brams and Davis, 'The 3/2's Rule in Presidential Campaigning', and S. J. Brams and M. D. Davis, 'Comment on "Campaign Resource Allocation under the Electoral College"' by C. S. Colantoni, T. J. Levesque and P. C. Ordeshook, American Political Science Review, 69 (1975), 155-6; Owen, 'Evaluation of a Presidential Election Game'.
54. C. S. Colantoni, T. J. Levesque and P. C. Ordeshook, 'Campaign Resource Allocation under the Electoral College' (with discussion), American Political Science Review, 69 (1975), 141-61.
55. W. S. Cleveland, 'Robust Locally Weighted Regression and Smoothing Scatterplots', Journal of the American Statistical Association, 74 (1979), 829-36.
56. Gelman, King and Boscardin, 'Estimating the Probability of Events That Have Never Occurred'.
57. Campbell, 'Forecasting the Presidential Vote in the States'.
58. The smallest states have slightly higher voting power, on average, which is attributable to the rule that all states, no matter how small, get at least three electoral votes.
59. C. B. Mulligan and C. G. Hunter, 'The Empirical Frequency of a Pivotal Vote', National Bureau of Economic Research Working Paper 8590.
60. See, for example, Gelman, King and Boscardin, 'Estimating the Probability of Events That Have Never Occurred'. See also B. I. Page and R. Y. Shapiro, The Rational Public: Fifty Years of Trends in Americans' Policy Preferences (Chicago: University of Chicago Press, 1992).
61. See, for example, Gelman, King and Boscardin, 'Estimating the Probability of Events That Have Never Occurred'. See also B. I. Page and R. Y. Shapiro, The Rational Public: Fifty Years of Trends in Americans' Policy Preferences (Chicago: University of Chicago Press, 1992).
62. A. Gelman, J. N. Katz, and F. Tuerlinckx, 'The Mathematics and Statistics of Voting Power', Statistical Science, 17 (2002), 420-35.
63. See P. Whittle, 'On the Variation of Yield Variance with Plot Size', Biometrika, 43 (1956), 337-43, for a general discussion of this statistical issue.
64. See D. J. Watts, P. S. Dodds and M. E. J. Newman, 'Identity and Search in Small-world Networks', Science, 296 (2002), 1302-15.
65. Penrose, 'The Elementary Statistics of Majority Voting'.
66. See, for example, Felsenthal and Machover, The Measurement of Voting Power, p. 12.
67. Snyder, Ting and Ansolabehere, 'Legislative Bargaining under Weighted Voting'.
68. Or 'parallel publics' in the terminology of Page and Shapiro, The Rational Public.
69. Leech, 'The Utility of the Voting Power Approach'.
70. For example, Leech notes that France and Germany have common interests in some European Union issues (Leech, 'The Utility of the Voting Power Approach').
71. As are those of Mulligan and Hunter, 'The Empirical Frequency of a Pivotal Vote'.
72. As in Felsenthal and Machover, 'Enlargement of the EU' and Leech, 'Designing the Voting System for the Council of the European Union'.
73. As in Felsenthal and Machover, 'Enlargement of the EU' and Leech, 'Designing the Voting System for the Council of the European Union'.
74. See also Good and Meyer, 'Estimating the Effcacy of a Vote', and H. Margolis, 'The Banzhaf Fallacy', American Journal of Political Science, 27 (1983), 321-6.
75. See also Good and Meyer, 'Estimating the Effcacy of a Vote', and H. Margolis, 'The Banzhaf Fallacy', American Journal of Political Science, 27 (1983), 321-6.